Computing Characterizations of Drugs for Ion Channels and Receptors Using Markov Models pp 109-118 | Cite as

# Computing Theoretical Drugs in the Two-Dimensional Case

## Abstract

*μ*; furthermore,

*μ*= 1 refers to the wild type case. Our aim is to devise a theoretical drug of the form

*k*

_{ bc },

*k*

_{ cb },

*k*

_{ bo }, and

*k*

_{ ob }are used to tune the drug such that the effect of the mutation is reduced as much as possible. As above, we will consider blockers associated with the closed state, which means that

*k*

_{ ob }= 0, or blockers associated with the open state, which means that

*k*

_{ cb }= 0. The model and discretization parameters used throughout this chapter are given in Table 6.1.

Parameters reused from the previous chapter (i.e., Table 5.1)

| 1 ms | |
---|---|---|

| 0.1 ms | |

| 0.01 ms | |

| 0.1 μM | |

| 1,000 μM | |

| 1 ms | |

| 1 ms | |

\(\Delta t\) | 0.001 ms | |

\(\Delta x\) | 0.92 μM | |

\(\Delta y\) | 9.3 μM |

## 6.1 Effect of the Mutation in the Two-Dimensional Case

*μ*= 1 (wild type) and

*μ*= 3 (mutant) and in Table 6.2 we give the statistics of the solutions. The total open probability increases from 0.430 for the wild type to 0.743 for the mutant. In addition, the expected concentrations of both the dyad and the junctional sarcoplasmic reticulum (JSR) decrease considerably. In the one-dimensional (1D) case we observed that the variability of the solution decreased when the mutation was introduced. This observation seems to carry over to the two-dimensional (2D) case.

Properties of the open probability density function in the wild type and mutant cases

Case | | \(E_{x_{o}}\) | \(E_{y_{o}}\) | \(\sigma _{x_{o}}\) | \(\sigma _{y_{o}}\) | |
---|---|---|---|---|---|---|

Wild type | 0.430 | 12.63 | 202.4 | 4.948 | 46.27 | |

Mutant | 0.743 | 9.64 | 131.7 | 2.419 | 18.90 |

## 6.2 A Closed State Drug

*k*

_{ bc }remains to be determined. To find the optimal value of this parameter, we need to extend the system (6.1) and (6.2) to account for the theoretical drug. When the closed state blocker is added, the steady state version of the probability density system reads

*k*

_{ bc }such that the open probability density function defined by the system (6.5)–(6.7) is as close as possible to the solution of the system (6.1) and (6.2) in the case of

*μ*= 1 (i.e., the wild type case). In other words, we want to use the drug to repair the effect of the mutations in the sense that we want the open probability densities to be as close as possible to the wild type open probability densities.

*μ*= 3 and

*k*

_{ bc }= 0. 01, 0.1, 1, and 10 ms

^{−1}. As expected, we note that the solution becomes increasingly similar to the wild type solution (see Fig. 6.1) as

*k*

_{ bc }increases.

### 6.2.1 Convergence as *k*_{bc} Increases

*k*

_{ bc }. To obtain a more precise impression of the convergence, we compute the norm of the difference between the open probability of the wild type case and the open probability of the solution of the system (6.5)–(6.7) as a function of

*k*

_{ bc }using the norm defined by ( 2.40) on page 46. The result is shown in Fig. 6.3 and we again observe that, when

*k*

_{ bc }becomes sufficiently large, the effect of the mutation is repaired completely.

## 6.3 An Open State Drug

*k*

_{ bo }and

*k*

_{ ob }.

### 6.3.1 Probability Density Model for Open State Blockers in 2D

*μ*= 1) and the open probability density function of the solution of the system (6.8)–(6.10) with

*μ*= 3. By minimizing the cost function, using Matlab’s

*Fminsearch*with default parameters and \(k_{ob} = k_{bo} = 1\) as an initial guess, we find that an optimal open state blocker is given by

#### 6.3.1.1 Does the Optimal Theoretical Drug Change with the Severity of the Mutation?

## 6.4 Statistical Properties of the Open and Closed State Blockers in 2D

We introduced statistical properties of probability density functions in Sect. 4.2 (see page 74). In Sect. 4.6 (page 90), we observed that, for the 1D release problem, the closed state blocker completely repaired the statistical properties of the open state probability density functions. In addition, an optimized version of an open state blocker gave good results, but it was unable to repair the standard deviation of the open state probability density functions for the particular CO-mutations we considered.

*k*

_{ bc }increases and the optimized version of the open state blocker also provides good results.

Statistical properties of the open probability density function in the mutant case when a blocker is applied. For the mutant case, we use *μ* = 3

Case | | \(E_{x_{o}}\) | \(E_{y_{o}}\) | \(\sigma _{x_{o}}\) | \(\sigma _{y_{o}}\) | |
---|---|---|---|---|---|---|

Closed blocker, | 0.547 | 10.55 | 144.2 | 4.726 | 58.93 | |

Closed blocker, | 0.465 | 13.60 | 188.9 | 5.890 | 73.66 | |

Closed blocker, | 0.422 | 13.69 | 205.7 | 5.231 | 53.08 | |

Closed blocker, | 0.428 | 12.80 | 203.2 | 5.014 | 47.15 | |

Open blocker, | 0.484 | 13.04 | 187.5 | 4.724 | 48.34 | |

Wild type | 0.430 | 12.63 | 202.4 | 4.948 | 46.27 | |

Mutant, no drug | 0.743 | 9.64 | 131.7 | 2.419 | 18.90 |

## 6.5 Numerical Comparison of Optimal Open and Closed State Blockers

*μ*= 1), the mutant (solution of (6.1) and (6.2) with

*μ*= 3), the optimal closed state blocker (solution of (6.5)–(6.7) using

*μ*= 3 and \(k_{bc} = 10\text{ ms}^{-1})\) and the optimal open state blocker (solution of (6.8)–(6.10) with \(\mu = 3,k_{ob} = 0.3225\text{ ms}^{-1},\) \(k_{bo} = 0.3346\text{ ms}^{-1}).\) We observe that it is hard to see any difference between the open probability density function of the wild type and the mutant when the closed state blocker is applied. In addition, the optimal open state blocker improves the solution, but not as much as the closed state blocker does.

## 6.6 Stochastic Simulations in 2D Using Optimal Drugs

*μ*= 3), and the mutant when the closed state blocker is applied (\(k_{bc} = 10\text{ ms}^{-1},\,k_{cb} = (\mu -1)k_{bc}\)). The dyad concentrations (

*x*=

*x*(

*t*)) are on the left-hand side and the JSR concentrations (

*y*=

*y*(

*t*)) are on the right-hand side. As for the 1D simulations, we observe that the mutations significantly reduce the variability of the solutions and that this effect is basically completely repaired by the closed state blocker.

## 6.7 Notes

- 1.
The 2D stochastic differential equation and the associated probability density system is taken from Huertas and Smith [35].

### References

- 35.M.A. Huertas, G.D. Smith, The dynamics of luminal depletion and the stochastic gating of Ca
^{2+}-activated Ca^{2+}channels and release sites. J. Theor. Biol.**246**(2), 332–354 (2007)MathSciNetCrossRefGoogle Scholar

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